{"title":"On an ErdŐs–Kac-Type Conjecture of Elliott","authors":"Ofir Gorodetsky, Lasse Grimmelt","doi":"10.1093/qmath/haae026","DOIUrl":null,"url":null,"abstract":"Elliott and Halberstam proved that $\\sum_{p \\lt n} 2^{\\omega(n-p)}$ is asymptotic to $\\phi(n)$. In analogy to the Erdős–Kac theorem, Elliott conjectured that if one restricts the summation to primes p such that $\\omega(n-p)\\le 2 \\log \\log n+\\lambda(2\\log \\log n)^{1/2}$ then the sum will be asymptotic to $\\phi(n)\\int_{-\\infty}^{\\lambda} \\mathrm{e}^{-t^2/2}\\,\\mathrm{d}t/\\sqrt{2\\pi}$. We show that this conjecture follows from the Bombieri–Vinogradov theorem. We further prove a related result involving Poisson–Dirichlet distribution, employing deeper lying level of distribution results of the primes.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/qmath/haae026","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Elliott and Halberstam proved that $\sum_{p \lt n} 2^{\omega(n-p)}$ is asymptotic to $\phi(n)$. In analogy to the Erdős–Kac theorem, Elliott conjectured that if one restricts the summation to primes p such that $\omega(n-p)\le 2 \log \log n+\lambda(2\log \log n)^{1/2}$ then the sum will be asymptotic to $\phi(n)\int_{-\infty}^{\lambda} \mathrm{e}^{-t^2/2}\,\mathrm{d}t/\sqrt{2\pi}$. We show that this conjecture follows from the Bombieri–Vinogradov theorem. We further prove a related result involving Poisson–Dirichlet distribution, employing deeper lying level of distribution results of the primes.
期刊介绍:
The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.